# Can the Derivative of a Function be Evaluated as n Approaches Infinity?

In summary, the conversation discusses the possibility of evaluating the derivative of a function of any order as n approaches infinity. It is mentioned that for certain functions, such as ex, the evaluation is trivial, but for others, such as cos(x), it cannot be done due to the cyclical nature of the function. The idea of using Cauchy's theorem and the generalized difference operator is brought up as potential methods for evaluation, but it is questioned whether they would actually work.
Let be a function so the derivative of any order n exist, my question is if there is a way to evaluate:

$$\frac{d^{n}f(x)}{dx^{n}}$$ as $$n\rightarrow \infty$$

Depends on the function. If it's ex, it's trivial. If it's a polynomial, it's trivial. If it's cos(x), it can't be done (cyclic, so there's no limit). If it's something that's a complete mess of a rational function, you're probably righteously too lazy to figure it out

Why should it exist some general way apart from differentiating the function?
The derivatives of the functions f(x)=cos(x) g(x)=1 "evolve" in totally different manners.

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well i was thinking about "Cauchy's theorem" so you have that the derivative of any order should satisfy:

$$2\pi i f^{(n)}(a)= n! \oint_{C}dzf(z)(z-a)^{-n-1}$$

and from this integral, if C is a circle of unit radius and centered at z=a then, the contour integral just becomes:

$$2\pi f^{(n)}(a)= n! \int_{-\pi}^{\pi} dxf(e^{ix}+a)e^{-inx}$$

EDIT: another good idea would be perhaps to use the "generalized difference" operator so..

$$f^{(n)}(x)=\frac{\nabla ^{n}}{h^{n}}$$ as h-->0 (small h)

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Why do you assume that Cauchy's theorem in general gives you a neat way to actually evaluate the derivative??

You are not Eljose, are you?

I'm supposing that f(z) is analytic with NO poles or at least that the point z=a is not a pole of f, then if f has no poles on the unit circle centered at z=a then "Cauchy Theorem" for derivatives holds.

I don't know what's this stuff about someone called eljose ..and what has to do with me or the forums...

## 1. What is the definition of a derivative?

A derivative is a mathematical concept that represents the rate of change or slope of a function at a particular point. It is often described as the instantaneous rate of change, as it measures the change in output with respect to a small change in the input.

## 2. What is the purpose of evaluating a derivative?

Evaluating a derivative allows us to analyze the behavior of a function at a specific point. It can help us determine the direction and steepness of a curve, as well as find maximum and minimum points on a graph. It is also essential in solving optimization problems in fields such as physics, economics, and engineering.

## 3. How is the derivative of a function calculated?

The derivative of a function is calculated using the limit definition of the derivative, also known as the difference quotient. This involves finding the slope of the secant line between two points on the function and then taking the limit as the two points get closer together. Alternatively, there are also rules and formulas for finding derivatives of common functions, such as the power rule, product rule, and chain rule.

## 4. Can the derivative of a function be negative?

Yes, the derivative of a function can be negative. This indicates that the function is decreasing at that particular point. For example, if the derivative of a position function is negative, it means that the object is moving in the negative direction with respect to time.

## 5. What is the difference between the derivative and integral of a function?

The derivative measures the rate of change of a function at a particular point, while the integral measures the accumulation of that function over a given interval. In other words, the derivative tells us how the function is changing, while the integral tells us how much of the function there is over a given area. They are inverse operations of each other, and the fundamental theorem of calculus connects them.

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